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Chapter 5: Q21E (page 212)

+\({{\rm{X}}_{\rm{1}}}{\rm{,}}{{\rm{X}}_{\rm{2}}}\)Question: Let, and \({{\rm{X}}_{\rm{3}}}\) be the lifetimes of components\({\rm{1,2}}\), and \({\rm{3}}\) in a three-component system.

a. How would you define the conditional pdf of \({{\rm{X}}_{\rm{3}}}\) given that \({{\rm{X}}_{\rm{1}}}{\rm{ = }}{{\rm{x}}_{\rm{1}}}\) and \({{\rm{X}}_{\rm{2}}}{\rm{ = }}{{\rm{x}}_{\rm{2}}}\) ?

b. How would you define the conditional joint pdf of \({{\rm{X}}_{\rm{2}}}\) and \({{\rm{X}}_{\rm{3}}}\) given that \({{\rm{X}}_{\rm{1}}}{\rm{ = }}{{\rm{x}}_{\rm{1}}}\) ?

Short Answer

Expert verified

a. The conditional pdf of \({{\rm{X}}_3}\) is \({{\rm{f}}_{{{\rm{X}}_{\rm{3}}}\mid {{\rm{X}}_{\rm{1}}}{\rm{,}}{{\rm{X}}_{\rm{2}}}}}\left( {{{\rm{x}}_{\rm{3}}}\mid {{\rm{x}}_{\rm{1}}}{\rm{,}}{{\rm{x}}_{\rm{2}}}} \right){\rm{ = }}\frac{{{\rm{f}}\left( {{{\rm{x}}_{\rm{1}}}{\rm{,}}{{\rm{x}}_{\rm{2}}}{\rm{,}}{{\rm{x}}_{\rm{3}}}} \right)}}{{{{\rm{f}}_{{{\rm{X}}_{\rm{1}}}{\rm{,}}{{\rm{X}}_{\rm{2}}}}}\left( {{{\rm{x}}_{\rm{1}}}{\rm{,}}{{\rm{x}}_{\rm{2}}}} \right)}}\).

b) The conditional pdf of \({{\rm{X}}_2}\) and \({{\rm{X}}_3}\) is \({{\rm{f}}_{{{\rm{X}}_{\rm{3}}}{\rm{,}}{{\rm{X}}_{\rm{2}}}\mid {{\rm{X}}_{\rm{1}}}}}\left( {{{\rm{x}}_{\rm{3}}}{\rm{,}}{{\rm{x}}_{\rm{2}}}\mid {{\rm{x}}_{\rm{1}}}} \right){\rm{ = }}\frac{{{\rm{f}}\left( {{{\rm{x}}_{\rm{1}}}{\rm{,}}{{\rm{x}}_{\rm{2}}}{\rm{,}}{{\rm{x}}_{\rm{3}}}} \right)}}{{{{\rm{f}}_{{{\rm{X}}_{\rm{1}}}}}\left( {{{\rm{x}}_{\rm{1}}}} \right)}}\).

Step by step solution

01

Definition

Probability simply refers to the likelihood of something occurring. We may talk about the probabilities of particular outcomes鈥攈ow likely they are鈥攚hen we're unclear about the result of an event. Statistics is the study of occurrences guided by probability.

02

Determining the conditional pdf of \({{\rm{X}}_{\rm{3}}}\)

(a):

The conditional probability density function of \({\rm{Y}}\) given that \({\rm{X = x}}\) is

1. \({{\rm{f}}_{{\rm{Y}}\mid {\rm{X}}}}{\rm{(y}}\mid {\rm{x) = }}\frac{{{\rm{f(x,y)}}}}{{{{\rm{f}}_{\rm{X}}}{\rm{(x)}}}}{\rm{,}}\;\;\;{\rm{ - 楼< y < 楼}}\)when \({\rm{X}}\) and \({\rm{Y}}\) are continuous rv's,

2. \({{\rm{p}}_{{\rm{Y}}\mid {\rm{X}}}}{\rm{(y}}\mid {\rm{x) = }}\frac{{{\rm{p(x,y)}}}}{{{{\rm{p}}_{\rm{X}}}{\rm{(x)}}}}{\rm{,}}\;\;\;{\rm{ - 楼< y < 楼}}\)when \({\rm{X}}\) and \({\rm{Y}}\) are discrete rv's.

Based on this, we define the conditional pdf of \({{\rm{X}}_{\rm{3}}}\) given that \({{\rm{X}}_{\rm{1}}}{\rm{ = }}{{\rm{x}}_{\rm{1}}}\) and $X_{2}=x_{2}$ as

\({{\rm{f}}_{{{\rm{X}}_{\rm{3}}}\mid {{\rm{x}}_{\rm{1}}}{\rm{,}}{{\rm{x}}_{\rm{2}}}}}\left( {{{\rm{x}}_{\rm{3}}}\mid {{\rm{x}}_{\rm{1}}}{\rm{,}}{{\rm{x}}_{\rm{2}}}} \right){\rm{ = }}\frac{{{\rm{f}}\left( {{{\rm{x}}_{\rm{1}}}{\rm{,}}{{\rm{x}}_{\rm{2}}}{\rm{,}}{{\rm{x}}_{\rm{3}}}} \right)}}{{\int {{{\rm{x}}_{\rm{1}}}} {\rm{,}}{{\rm{x}}_{\rm{2}}}\left( {{{\rm{x}}_{\rm{1}}}{\rm{,}}{{\rm{x}}_{\rm{2}}}} \right)}}\)

where \({\rm{f}}\left( {{{\rm{x}}_{\rm{1}}}{\rm{,}}{{\rm{x}}_{\rm{2}}}{\rm{,}}{{\rm{x}}_{\rm{3}}}} \right)\) is joint pdf of\({{\rm{X}}_{\rm{1}}}{\rm{,}}{{\rm{X}}_{\rm{2}}}\), and\({{\rm{X}}_{\rm{3}}}\), and \({{\rm{f}}_{{{\rm{X}}_{\rm{1}}}{\rm{,}}{{\rm{X}}_{\rm{2}}}}}\left( {{{\rm{x}}_{\rm{1}}}{\rm{,}}{{\rm{x}}_{\rm{2}}}} \right)\) is marginal joint pdf of\({{\rm{X}}_{\rm{1}}}\), and\({{\rm{X}}_{\rm{2}}}\).

03

Determining the conditional joint pdf of \({{\rm{X}}_{\rm{2}}}\) and \({{\rm{X}}_{\rm{3}}}\)

(b):

We define the conditional joint pdf of\({{\rm{X}}_{\rm{2}}}\), and \({{\rm{X}}_{\rm{3}}}\) given that \({{\rm{X}}_{\rm{1}}}{\rm{ = }}{{\rm{x}}_{\rm{1}}}\) as follows

\({{\rm{f}}_{{{\rm{X}}_{\rm{3}}}{\rm{,}}{{\rm{X}}_{\rm{2}}}\mid {{\rm{X}}_{\rm{1}}}}}\left( {{{\rm{x}}_{\rm{3}}}{\rm{,}}{{\rm{x}}_{\rm{2}}}\mid {{\rm{x}}_{\rm{1}}}} \right){\rm{ = }}\frac{{{\rm{f}}\left( {{{\rm{x}}_{\rm{1}}}{\rm{,}}{{\rm{x}}_{\rm{2}}}{\rm{,}}{{\rm{x}}_{\rm{3}}}} \right)}}{{{{\rm{f}}_{{{\rm{X}}_{\rm{1}}}}}\left( {{{\rm{x}}_{\rm{1}}}} \right)}}{\rm{,}}\)

where \({\rm{f}}\left( {{{\rm{x}}_{\rm{1}}}{\rm{,}}{{\rm{x}}_{\rm{2}}}{\rm{,}}{{\rm{x}}_{\rm{3}}}} \right)\) is joint pdf of\({{\rm{X}}_{\rm{1}}}{\rm{,}}{{\rm{X}}_{\rm{2}}}\), and\({{\rm{X}}_{\rm{3}}}\), and \({{\rm{f}}_{{{\rm{X}}_{\rm{1}}}}}\left( {{{\rm{x}}_{\rm{1}}}} \right)\) is marginal pdf of\({{\rm{X}}_{\rm{1}}}\).

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Most popular questions from this chapter

Let \({{\rm{X}}_{\rm{1}}}{\rm{,}}{{\rm{X}}_{\rm{2}}}{\rm{, \ldots ,}}{{\rm{X}}_{\rm{n}}}\)be random variables denoting \({\rm{X}}\)independent bids for an item that is for sale. Suppose each \({\rm{X}}\)is uniformly distributed on the interval \({\rm{(100,200)}}\).If the seller sells to the highest bidder, how much can he expect to earn on the sale? (Hint: Let \({\rm{Y = max}}\left( {{{\rm{X}}_{\rm{1}}}{\rm{,}}{{\rm{X}}_{\rm{2}}}{\rm{, \ldots ,}}{{\rm{X}}_{\rm{n}}}} \right)\).First find \({{\rm{F}}_{\rm{Y}}}{\rm{(y)}}\)by noting that \({\rm{Y}}\)iff each \({{\rm{X}}_{\rm{i}}}\)is \({\rm{y}}\). Then obtain the pdf and \({\rm{E(Y)}}\).

A company maintains three offices in a certain region, each staffed by two employees. Information concerning yearly salaries (\({\rm{1000}}\)s of dollars) is as follows:

\(\begin{array}{*{20}{c}}{{\rm{ Office }}}&{\rm{1}}&{\rm{1}}&{\rm{2}}&{\rm{2}}&{\rm{3}}&{\rm{3}}\\{{\rm{ Employee }}}&{\rm{1}}&{\rm{2}}&{\rm{3}}&{\rm{4}}&{\rm{5}}&{\rm{6}}\\{{\rm{ Salary }}}&{{\rm{29}}{\rm{.7}}}&{{\rm{33}}{\rm{.6}}}&{{\rm{30}}{\rm{.2}}}&{{\rm{33}}{\rm{.6}}}&{{\rm{25}}{\rm{.8}}}&{{\rm{29}}{\rm{.7}}}\end{array}\)

a. Suppose two of these employees are randomly selected from among the six (without replacement). Determine the sampling distribution of the sample mean salary\({\rm{\bar X}}\).

b. Suppose one of the three offices is randomly selected. Let\({{\rm{X}}_{\rm{1}}}\)and\({{\rm{X}}_{\rm{2}}}\)denote the salaries of the two employees. Determine the sampling distribution of\({\rm{\bar X}}\).

c. How does \({\rm{E(\bar X)}}\) from parts (a) and (b) compare to the population mean salary\({\rm{\mu }}\)?

a. Let \({{\rm{X}}_{\rm{1}}}\)have a chi-squared distribution with parameter \({{\rm{\nu }}_{\rm{1}}}\) (see Section 4.4), and let \({{\rm{X}}_{\rm{2}}}\)be independent of \({{\rm{X}}_{\rm{1}}}\)and have a chi-squared distribution with parameter\({{\rm{v}}_{\rm{2}}}\). Use the technique of to show that \({{\rm{X}}_{\rm{1}}}{\rm{ + }}{{\rm{X}}_{\rm{2}}}\)has a chi-squared distribution with parameter\({{\rm{v}}_{\rm{1}}}{\rm{ + }}{{\rm{v}}_{\rm{2}}}\).

b. You were asked to show that if \({\rm{Z}}\)is a standard normal \({\rm{rv}}\), then \({{\rm{Z}}^{\rm{2}}}\)has a chi squared distribution with\({\rm{v = 1}}\). Let \({{\rm{Z}}_{\rm{1}}}{\rm{,}}{{\rm{Z}}_{\rm{2}}}{\rm{, \ldots ,}}{{\rm{Z}}_{\rm{n}}}\)be \({\rm{n}}\)independent standard normal \({\rm{rv}}\) 's. What is the distribution of\({\rm{Z}}_{\rm{1}}^{\rm{2}}{\rm{ + }}...{\rm{ + Z}}_{\rm{n}}^{\rm{2}}\)? Justify your answer.

c. Let \({{\rm{X}}_{\rm{1}}}{\rm{, \ldots ,}}{{\rm{X}}_{\rm{n}}}\)be a random sample from a normal distribution with mean \({\rm{\mu }}\)and variance\({{\rm{\sigma }}^{\rm{2}}}\). What is the distribution of the sum \({\rm{Y = }}\sum\limits_{{\rm{i = 1}}}^{\rm{n}} {{{\left( {\left( {{{\rm{X}}_{\rm{i}}}{\rm{ - \mu }}} \right){\rm{/\sigma }}} \right)}^{\rm{2}}}} {\rm{?}}\)Justify your answer?

Two components of a minicomputer have the following joint pdf for their useful lifetimes \({\rm{X}}\)and \({\rm{Y}}\)

a. What is the probability that the lifetime \({\rm{X}}\) of the first component exceeds \({\rm{3}}\)?

b. What are the marginal pdf鈥檚 of \({\rm{X}}\)and \({\rm{Y}}\)? Are the two lifetimes independent? Explain.

c. What is the probability that the lifetime of at least one component exceeds\({\rm{3}}\)?

The National Health Statistics Reports dated Oct. \({\rm{22, 2008}}\), stated that for a sample size of \({\rm{277 18 - }}\)year-old American males, the sample mean waist circumference was \({\rm{86}}{\rm{.3cm}}\). A somewhat complicated method was used to estimate various population percentiles, resulting in the following values:

a. Is it plausible that the waist size distribution is at least approximately normal? Explain your reasoning. If your answer is no, conjecture the shape of the population distribution.

b. Suppose that the population mean waist size is \({\rm{85cm}}\)and that the population standard deviation is \({\rm{15cm}}\). How likely is it that a random sample of \({\rm{277}}\) individuals will result in a sample mean waist size of at least \({\rm{86}}{\rm{.3cm}}\)?

c. Referring back to (b), suppose now that the population mean waist size in \({\rm{82cm}}\).Now what is the (approximate) probability that the sample mean will be at least \({\rm{86}}{\rm{.3cm}}\)? In light of this calculation, do you think that \({\rm{82cm}}\)is a reasonable value for \({\rm{\mu }}\)?
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